Solve d/dxleft(3left(2xright)^2-2/9x-5right)

$$\frac{d}{d x } \left( \frac{ 3 { \left(2x \right) }^{ 2 } -2 }{ 9x-5 } \right)$$

Evaluate

$$\frac{6\left(18x^{2}-20x+3\right)}{\left(9x-5\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Expand \left(2x\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\times 2^{2}x^{2}-2}{9x-5})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\times 4x^{2}-2}{9x-5})$$

Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{12x^{2}-2}{9x-5})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(9x^{1}-5\right)\frac{\mathrm{d}}{\mathrm{d}x}(12x^{2}-2)-\left(12x^{2}-2\right)\frac{\mathrm{d}}{\mathrm{d}x}(9x^{1}-5)}{\left(9x^{1}-5\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{\left(9x^{1}-5\right)\times 2\times 12x^{2-1}-\left(12x^{2}-2\right)\times 9x^{1-1}}{\left(9x^{1}-5\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(9x^{1}-5\right)\times 24x^{1}-\left(12x^{2}-2\right)\times 9x^{0}}{\left(9x^{1}-5\right)^{2}}$$

Expand using distributive property.

$$\frac{9x^{1}\times 24x^{1}-5\times 24x^{1}-\left(12x^{2}\times 9x^{0}-2\times 9x^{0}\right)}{\left(9x^{1}-5\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{9\times 24x^{1+1}-5\times 24x^{1}-\left(12\times 9x^{2}-2\times 9x^{0}\right)}{\left(9x^{1}-5\right)^{2}}$$

Do the arithmetic.

$$\frac{216x^{2}-120x^{1}-\left(108x^{2}-18x^{0}\right)}{\left(9x^{1}-5\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{216x^{2}-120x^{1}-108x^{2}-\left(-18x^{0}\right)}{\left(9x^{1}-5\right)^{2}}$$

Combine like terms.

$$\frac{\left(216-108\right)x^{2}-120x^{1}-\left(-18x^{0}\right)}{\left(9x^{1}-5\right)^{2}}$$

Subtract 108 from 216.

$$\frac{108x^{2}-120x^{1}-\left(-18x^{0}\right)}{\left(9x^{1}-5\right)^{2}}$$

For any term t, t^{1}=t.

$$\frac{108x^{2}-120x-\left(-18x^{0}\right)}{\left(9x-5\right)^{2}}$$

For any term t except 0, t^{0}=1.

$$\frac{108x^{2}-120x-\left(-18\right)}{\left(9x-5\right)^{2}}$$

Differentiate w.r.t. x

$$\frac{276}{\left(9x-5\right)^{3}}$$